Lee Johnson has also been hit with a £3,000 fine and a four-match touchline ban for his role in the scenes, which he will have to serve in his next role.
Lincoln City have been £4,000 after both clubs admitted to a charge of failing to ensure their players conducted themselves in an orderly fashion, and failing to refrain from provocative behaviour in the 95th minute.
The game had been bad-tempered as Chris Maguire scored a hat-trick on his return to the Stadium of Light, and the confrontation began when Callum Doyle was booked for a rash challenge near the Sunderland dug-out.
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Johnson requested a personal hearing to argue that he was acting in defence of his young player, but the FA have dismissed those claims, finding his behaviour in the initial incident and following his subsequent red card to be improper.
Speaking immediately after the game, the former Sunderland head coach had said: “I thought it was a high tackle from Callum (Doyle). It wasn’t the best tackle in the world but obviously when there is an 18-year-old lad, I know he’s a big lad, getting picked on right in front of me, I felt it was right to try and prevent and defend.
“The craziness is I get sent off and obviously nobody else does. That’s frustrating but I’ve got to be accountable. I will be and I’ll take any punishment on the chin.”